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NEC says "Where circuit conductors are spliced within a box, or terminated on equipment within or supported by a box, any equipment grounding conductor(s) associated with those circuit conductors shall be connected within the box or to the box with devices suitable for the use ..." 250.148.

Even better, in case someone feels that a panelboard enclosure isn't a box , is 408.40. "... The terminal bar shall be bonded to the cabinet and panelboard frame, if of metal, ..."

What does code say about the need of bonding the equipment ground bar to the chassis of the breaker box on a sub-panel, that is connected to the main with EMT conduit?

If this is an equipment ground bar then the screws that are provided with the accessory will suffice to bond the bar to the box. If you are using EMT then there is no need to include an egc-- if that is what you are asking.